23.7 Sampling difficulties with problematic priors
With an improper posterior, it is theoretically impossible to properly explore the posterior. However, Gibbs sampling as performed by BUGS and JAGS, although still unable to properly sample from such an improper posterior, behaves differently in practice than the Hamiltonian Monte Carlo sampling performed by Stan when faced with an example such as the two intercept model discussed in the collinearity section and illustrated in the non-identifiable density plot.
Gibbs sampling
Gibbs sampling, as performed by BUGS and JAGS, may appear to be efficient and well behaved for this unidentified model, but as discussed in the previous subsection, will not actually explore the posterior properly.
Consider what happens with initial values \(\lambda_1^{(0)}, \lambda_2^{(0)}\). Gibbs sampling proceeds in iteration \(m\) by drawing \[\begin{align*} \lambda_1^{(m)} &\sim p(\lambda_1 \mid \lambda_2^{(m-1)}, \sigma^{(m-1)}, y) \\ \lambda_2^{(m)} &\sim p(\lambda_2 \mid \lambda_1^{(m)}, \sigma^{(m-1)}, y) \\ \sigma^{(m)} &\sim p(\sigma \mid \lambda_1^{(m)}, \lambda_2^{(m)}, y). \end{align*}\]
Now consider the draw for \(\lambda_1\) (the draw for \(\lambda_2\) is symmetric), which is conjugate in this model and thus can be done efficiently. In this model, the range from which the next \(\lambda_1\) can be drawn is highly constrained by the current values of \(\lambda_2\) and \(\sigma\). Gibbs will run quickly and provide seemingly reasonable inferences for \(\lambda_1 + \lambda_2\). But it will not explore the full range of the posterior; it will merely take a slow random walk from the initial values. This random walk behavior is typical of Gibbs sampling when posteriors are highly correlated and the primary reason to prefer Hamiltonian Monte Carlo to Gibbs sampling for models with parameters correlated in the posterior.
Hamiltonian Monte Carlo sampling
Hamiltonian Monte Carlo (HMC), as performed by Stan, is much more efficient at exploring posteriors in models where parameters are correlated in the posterior. In this particular example, the Hamiltonian dynamics (i.e., the motion of a fictitious particle given random momentum in the field defined by the negative log posterior) is going to run up and down along the valley defined by the potential energy (ridges in log posteriors correspond to valleys in potential energy). In practice, even with a random momentum for \(\lambda_1\) and \(\lambda_2\), the gradient of the log posterior is going to adjust for the correlation and the simulation will run \(\lambda_1\) and \(\lambda_2\) in opposite directions along the valley corresponding to the ridge in the posterior log density.
No-U-turn sampling
Stan’s default no-U-turn sampler (NUTS), is even more efficient at exploring the posterior (see Hoffman and Gelman 2014). NUTS simulates the motion of the fictitious particle representing the parameter values until it makes a U-turn, it will be defeated in most cases, as it will just move down the potential energy valley indefinitely without making a U-turn. What happens in practice is that the maximum number of leapfrog steps in the simulation will be hit in many of the iterations, causing a large number of log probability and gradient evaluations (1000 if the max tree depth is set to 10, as in the default). Thus sampling will appear to be slow. This is indicative of an improper posterior, not a bug in the NUTS algorithm or its implementation. It is simply not possible to sample from an improper posterior! Thus the behavior of HMC in general and NUTS in particular should be reassuring in that it will clearly fail in cases of improper posteriors, resulting in a clean diagnostic of sweeping out large paths in the posterior.
Here are results of Stan runs with default parameters fit to \(N=100\) data points generated from \(y_n \sim \textsf{normal}(0,1)\):
Two Scale Parameters, Improper Prior
Inference for Stan model: improper_stan
Warmup took (2.7, 2.6, 2.9, 2.9) seconds, 11 seconds total
Sampling took (3.4, 3.7, 3.6, 3.4) seconds, 14 seconds total
Mean MCSE StdDev 5% 95% N_Eff N_Eff/s R_hat
lp__ -5.3e+01 7.0e-02 8.5e-01 -5.5e+01 -5.3e+01 150 11 1.0
n_leapfrog__ 1.4e+03 1.7e+01 9.2e+02 3.0e+00 2.0e+03 2987 212 1.0
lambda1 1.3e+03 1.9e+03 2.7e+03 -2.3e+03 6.0e+03 2.1 0.15 5.2
lambda2 -1.3e+03 1.9e+03 2.7e+03 -6.0e+03 2.3e+03 2.1 0.15 5.2
sigma 1.0e+00 8.5e-03 6.2e-02 9.5e-01 1.2e+00 54 3.9 1.1
mu 1.6e-01 1.9e-03 1.0e-01 -8.3e-03 3.3e-01 2966 211 1.0Two Scale Parameters, Weak Prior
Warmup took (0.40, 0.44, 0.40, 0.36) seconds, 1.6 seconds total
Sampling took (0.47, 0.40, 0.47, 0.39) seconds, 1.7 seconds total
Mean MCSE StdDev 5% 95% N_Eff N_Eff/s R_hat
lp__ -54 4.9e-02 1.3e+00 -5.7e+01 -53 728 421 1.0
n_leapfrog__ 157 2.8e+00 1.5e+02 3.0e+00 511 3085 1784 1.0
lambda1 0.31 2.8e-01 7.1e+00 -1.2e+01 12 638 369 1.0
lambda2 -0.14 2.8e-01 7.1e+00 -1.2e+01 12 638 369 1.0
sigma 1.0 2.6e-03 8.0e-02 9.2e-01 1.2 939 543 1.0
mu 0.16 1.8e-03 1.0e-01 -8.1e-03 0.33 3289 1902 1.0One Scale Parameter, Improper Prior
Warmup took (0.011, 0.012, 0.011, 0.011) seconds, 0.044 seconds total
Sampling took (0.017, 0.020, 0.020, 0.019) seconds, 0.077 seconds total
Mean MCSE StdDev 5% 50% 95% N_Eff N_Eff/s R_hat
lp__ -54 2.5e-02 0.91 -5.5e+01 -53 -53 1318 17198 1.0
n_leapfrog__ 3.2 2.7e-01 1.7 1.0e+00 3.0 7.0 39 507 1.0
mu 0.17 2.1e-03 0.10 -3.8e-03 0.17 0.33 2408 31417 1.0
sigma 1.0 1.6e-03 0.071 9.3e-01 1.0 1.2 2094 27321 1.0On the top is the non-identified model with improper uniform priors and likelihood \(y_n \sim \textsf{normal}(\lambda_1 + \lambda_2, \sigma)\).
In the middle is the same likelihood as the middle plus priors \(\lambda_k \sim \textsf{normal}(0,10)\).
On the bottom is an identified model with an improper prior, with likelihood \(y_n \sim \textsf{normal}(\mu,\sigma)\). All models estimate \(\mu\) at roughly 0.16 with low Monte Carlo standard error, but a high posterior standard deviation of 0.1; the true value \(\mu=0\) is within the 90% posterior intervals in all three models.
Examples: fits in Stan
To illustrate the issues with sampling from non-identified and only weakly identified models, we fit three models with increasing degrees of identification of their parameters. The posteriors for these models is illustrated in the non-identifiable density plot. The first model is the unidentified model with two location parameters and no priors discussed in the collinearity section.
data {
int N;
array[N] real y;
}
parameters {
real lambda1;
real lambda2;
real<lower=0> sigma;
}
transformed parameters {
real mu;
mu = lambda1 + lambda2;
}
model {
y ~ normal(mu, sigma);
}The second adds priors to the model block for lambda1 and
lambda2 to the previous model.
lambda1 ~ normal(0, 10);
lambda2 ~ normal(0, 10);The third involves a single location parameter, but no priors.
data {
int N;
array[N] real y;
}
parameters {
real mu;
real<lower=0> sigma;
}
model {
y ~ normal(mu, sigma);
}All three of the example models were fit in Stan 2.1.0 with default parameters (1000 warmup iterations, 1000 sampling iterations, NUTS sampler with max tree depth of 10). The results are shown in the non-identified fits figure. The key statistics from these outputs are the following.
As indicated by
R_hatcolumn, all parameters have converged other than \(\lambda_1\) and \(\lambda_2\) in the non-identified model.The average number of leapfrog steps is roughly 3 in the identified model, 150 in the model identified by a weak prior, and 1400 in the non-identified model.
The number of effective samples per second for \(\mu\) is roughly 31,000 in the identified model, 1,900 in the model identified with weakly informative priors, and 200 in the non-identified model; the results are similar for \(\sigma\).
In the non-identified model, the 95% interval for \(\lambda_1\) is (-2300,6000), whereas it is only (-12,12) in the model identified with weakly informative priors.
In all three models, the simulated value of \(\mu=0\) and \(\sigma=1\) are well within the posterior 90% intervals.
The first two points, lack of convergence and hitting the maximum number of leapfrog steps (equivalently maximum tree depth) are indicative of improper posteriors. Thus rather than covering up the problem with poor sampling as may be done with Gibbs samplers, Hamiltonian Monte Carlo tries to explore the posterior and its failure is a clear indication that something is amiss in the model.